Rigidity of higher dimensional Lie groups -- Toward comprehensive understanding

2020 Fiscal Year Final Research Report

• Project/Area Number: 18K03294
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: The University of Tokyo
• Principal Investigator: Kanai Masahiko 東京大学, 大学院数理科学研究科, 教授 (70183035)
• Project Period (FY): 2018-04-01 – 2021-03-31
• Keywords: 剛性 / 高階リー群 / トンプソン群

Outline of Final Research Achievements

It is known that semi-simple Lie groups of real-rank greater than or equal to 2 exhibit rigidity phenomena. The proofs of such results are often done by appealing to a classification of such Lie groups (or their Lie algebras). The present research aims at unifying those theorems. I gave a new proof of Matsushima's vanishing based on such a scenario. A basic strategy is -- For a certain foliated space, apply the theory of harmonic integration in the direction tangent to the foliation, and ergodic theory in the transverse direction.
We made investigations on the Thompson group F, as well. In particular, I discovered a deep similarity between a result on the automorphism group of F done by Brin et al., and a new infinite-dimensial space on which F acts.

Free Research Field

Geometry

Academic Significance and Societal Importance of the Research Achievements

最終年度にコロナ禍に襲われたこともあり，本研究計画は完成にはほど遠いと言わざるをえない．しかし，すでに部分的な結果は得られている．それらを発展させ，さらに新たな考察を積み上げれば，この分野の専門家たちから十分に高い評価を得られるのではないかと期待している．